Render plane only where spotlight hits, rest should be transparent
use the color of the non-lit plane for the 'Key Color' attribute.

this makes everything transparent that's with the same color. if your logo also gets transparent then just wait a bit, i'll create a more complicated but better working compositor for you.

Edit: here's the better compositor

for this one you need to also set the plane's pass index to 1

and check the render layer's object index

I can't add more than 2 images so you need to use google for the last part